CLASSICAL TIME SERIES DECOMPOSITION

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "CLASSICAL TIME SERIES DECOMPOSITION"

Transcription

1 Time Series Lecure Noes, MSc in Operaional Research Lecure CLASSICAL TIME SERIES DECOMPOSITION Inroducion We menioned in lecure ha afer we calculaed he rend, everyhing else ha remained (according o ha calculaion approach) was called he residuals. We also saed ha here are oher mehods ha give paricular imporance o hese residuals'. These mehods rea his remainder as a valuable componen of he ime series, and he basis for using hese mehods is o find he componens ha make up he series. Le us develop furher our iniial formula according o which: Time series = Paern + Residuals If we give a name o he paern and call i a rend, hen he residuals could be called variaions around he rend. The variaions could be of a differen naure and we can differeniae he following variaions: Cyclical variaions Seasonal variaions Irregular variaions In he pas, when his mehod was developed for census purposes, rend was called a `secular variaion' or a `secular endency'. Almos a hundred years ago, saisicians developed a mehod ha we, call oday a classical decomposiion mehod, and he basic assumpion of his mehod was ha each ime series consised of he following componens: T = rend C = cyclical componen, S = seasonal componen, I = irregular variaions Decomposiion echniques are among he oldes of he forecasing mehods. Economiss have used hese echniques since he beginning of he cenury o idenify he business cycle. Decomposiion mehods are among he easies o undersand and use, especially for shor-erm forecasing. In addiion, unforunaely, he decomposiion mehod is basically inuiive and here are number of heoreical weaknesses in is approach. However, hese do no preven he posiive resuls obained in he pracical applicaion of he mehod. Addiive and Muliplicaive Models Time-series models can basically be classified ino wo ypes: addiive model and muliplicaive models. For an addiive model, we assume ha he daa is he sum of he ime-series componens, ha is,. X = T + C + S + I + e. X = T C S I + e If he daa does no conain one of he componens, he value for ha componen is equal o zero. In an addiive model he seasonal or cyclical componen is independen of he rend, and hus he magniude of he seasonal swing (movemen) is consan over he ime, as illusraed in Fig. (a). Figure (a): addiive model-he magniude of he seasonal swing is consan over ime.

2 Time Series Lecure Noes, Figure (b) muliplicaive model he magniude of he seasonal swing is proporional o he rend. In a muliplicaive model, he daa is he produc of he various componens, ha is, X = T S C e, If rend, seasonal variaion, or cycle is missing, hen is value is assumed o be. As shown in Fig. b, he seasonal (or cyclical) facor of a muliplicaive model is proporional (a raio) o he rend, and hus he magniude of he seasonal swing increases or decreases according o he behavior of he rend. Alhough mos daa ha possess seasonal (cyclical) variaions canno be precisely classified as addiive or muliplicaive in naure, we usually look a he forecass obained using boh models and choose he model ha yields he smalles SSE and ha seems appropriae for he daa in quesion. The Seasonal and Cyclical Componens Daa ha is repored quarerly, monhly, weekly, ec. and ha demonsraes a yearly periodic paern is said o conain a seasonal componen or facor. A seasonal series may be rended or unrended. I may or may no possess a cyclical componen. However, in mos cases seasonaliy is easier o model han rend or cycle because i has a clearly repeiive -monh or -quarer paern. Trend may be linear or curvilinear, cycles can be any lengh and may repea a irregular inervals, bu seasonaliy is usually well defined. In he decomposiion mehod, he seasonal componen is he firs componen ha is modeled in he ime series. Addiive Decomposiion The addiive decomposiion mehod is appropriae for modeling ime-series daa conaining rend, seasonal, and error componens, if we can assume he following: We have an addiive model ( X = T + S + C + e ), he error erms are random, and he seasonal componen for any one season is he same in each year. Seps in he Decomposiion Mehod Figure Quarerly sales figures for he years 985 hrough 988 Figure displays quarerly sales figures for he years 985 hrough 988. Clearly, sales of he company is seasonal. There is a definable drop in sales during he firs quarer and a rise o a peak during he hird quarer. There also appears o be an upward rend in he daa. We assume ha we have an addiive model and use his daa o explain and demonsrae he following seps in he addiive decomposiion mehod. To accomplish his, cenered moving averages (wo-period moving averages of he iniial moving averages) are compued:

3 Time Series Lecure Noes, cenered moving average (CMA,) = rend + cycle, Table Obaining he esimaes for seasonaliy and in an addiive decomposiion model of consrucion sales X ( ) Year Quarer T X Moving CMA S + e S d Ave; T C As shown in he Table,. firs moving average = ( ) / = 7. 6 second moving average = ( ) / = hird moving average = ( ) / = 6. ec. Thus, CMA = ( ) / = 5. CMA = ( ) / = 58. CMA 5 = ( ) / = 6.9. Subrac he CMA (T + C ) from he daa. The difference is equal o S + e In he example, S + e = = 0.56 S + e = = 7.56 S5 + e5 = = 8.9 ec.. Remove he error ( e ) componen from S + e by compuing he average for each of he seasons. Tha is, Table Seasonal values of daa series is given in he Table. Quarer Quarer Quarer Quarer S =-0.5 S =5.0 S =.5 S =5. These are he four esimaes for he seasonal componens.. These averaged seasonal esimaes should add up o zero. If hey do no, we mus adjus hem (normalize hem) so ha hey will be final adjusmen (normalizaion) consiss of subracing a consan ( Sn / L) from each esimae. Using he daa from Table, ( Sn / L) = ( ) / = 0. he final seasonal esimaes are = = 0.86 S = =. 96 S

4 Time Series Lecure Noes, S =.5 0. =. S = = Deseasonlize he daa by subracing from i heir proper seasonal esimaes: di = x i Sn For example, d = 6 ( 0.86) = 6.86 d = 6.8 (.96) =.8 d = 6.9 (.) = d = 65.7 (.79) = 60.9 d 6 = 50.8 (.79) = Perform he proper regression analysis on he deseasonlized daa o obain he appropriae model for he rend. The appropriae model for he daa in Table is a linear model ( - es for he slope =.). The equaion o he model he rend is T = An esimae or forecas for any ime period can be found by adding ogeher he esimaes for he various componens. For our example, he forecas for he sevenh ime period would be Xˆ = T7 + S7 + C7, T 7 = (7) = S7 = S =. C 7 = 0 (we are assuming here is no cycle) Xˆ = = 79.5 Figure Graph of acual and esimaed values of daa in he Table... Forecas and Confidence Inervals we discussed he mehods of forecasing ime series daa an addiive model. The procedure was o add he appropriae esimaes for he various componens ogeher. Thus, he forecas conrucing for he firs quarer of 989 should be Xˆ 7 = T7 + S7 T 7 = (7) = S 7 = S = X 7 = (-0.86) = 90.5 Since he decomposiion mehod is basically inuiive, wihou any sound saisical heory behind i, here is no "saisically correc" confidence inerval for x. However, here is an inuiive mehod for consrucing confidence inervals for decomposiion forecass. The mehod is simply o use he inerval error for he rend model as he measure of he inerval error for x. This can be compued from he resuls of he regression analysis on he deseasonalized daa. Therefore, he confidence inerval for xˆ, is compued as Xˆ ± α / Se (correcion facor) where S e = he sandard error of esimae ( MSE ) from he appropriae rend regression analysis, ( p ) and where correcion facor = + + where p is he ph poin. n ( ) n which is obained from he resuls of he regression analysis of he deseasonalized daa.

5 Time Series Lecure Noes, 5 In our example, he rend sandard error for he deseasonalized daa is equal o 5.86, and he correcion facor for he linear model is equal o.9. Thus, an approximae 95 percen confidence inerval for X 7 is 90.5 ±.5 x (5.86) x(.9), or (75.9 o 50.) We should sress ha his echnique yields only an approximae (alhough fairly accurae) confidence inerval and does no have a sound heoreical basis. Muliplicaive Decomposiion The muliplicaive decomposiion mehod (someimes called he raio o rend or he raio o moving average) is very similar o ha of he addiive decomposiion mehod. For a muliplicaive decomposiion model, we assume he following: X, is a produc of he various componens, including error ( X = TSC e ) erms are random and he seasonal facor for any one season is he same for each year. Seps in he Decomposiion Mehod To illusrae he echniques used in he muliplicaive decomposiion mehod, we will use U.S. quarerly reail sales daa for he years As seen in Fig., and he daa is in he Table, here is a regular seasonal paern in he series. Pronounced peaks during he fourh quarer and hrough during he firs quarer of each year are apparen. There also appears o be an upward rend in he daa. Thus, we proceed o isolae hese componens by applying he following seps o he daa: Sales(millions of dollars) Acvual values of U.S reail sales(98-987) Qurer Figure : Acual values are shown in he Table for U.S reail sales (98-987). For he acual ime series, compue (as in he addiive model) a cenered moving average of lengh L. The moving average and cenered moving average for he firs hree quarers of he reail sales daa (Table ) are Quarerly daa (in millions of dollars) was obained by summing monhly daa. 00 for compuaional convenience divided he new series. Linear regression resuls a = b = 8.79 es for slope = 7.97, r = compued in he following manner: firs moving average = (,88 +,9 +,80 +,505)/ =,0.75 second moving average = (,9 +,80 +,505 +,00)/ =,8.5 hird moving average = (,80 +,505 +,00 +,9)/ =,88.5 ec. and CMA = (,0.75 +,8.5)/ =,.5 CMA = (,8.5 +,88.5)/ =,6.5 CMA 5 = (, ,6.5) / =,5 (see Table.) ec.

6 Time Series Lecure Noes, 6 Table : Obaining he esimaes for seasonaliy and rend in a muliplicaive decomposiion model of reail sales in (99-997) Year Quarer T x Sales Moving Ave; CMA T C S e S Divide he CMA, ( T C ) ino he daa. The quoien is equal o S ne : X = T S C e ( TSC e ) (TC ) = Se In our example S e = 80.5 = e = S = S5 e5 = 00 5 = ec: d Remove he error ( e ) componen from S ne by compuing he average for each of he seasons: Table Seasonal values of daa series is given in he Table Quarer Quarer Quarer Quarer S =0.905 S =.0 S =.00 S =.07 These are he four esimaes for he seasonal componens.. These averaged seasonal esimaes should, for a muliplicaive model, add up L (he number of seasons in a year). If hey do no, we mus normalize hem so ha hey will. The final normalizaion consiss of muliplying each esimae by he consan L ( Sn ). Using he daa fin he Table, L ( Sn ) = /( ) =.005 The final seasonal esimaes are S = = 0.9 S = =. 05 S = =.005 S = = Deseasonalize he daa by dividing i by he proper seasonal esimaes: d = X S, For example, d, =,88/ 0.90 =,86.97 d =,9 /.05 =, d =,80 /.005 =,6.79 d =,505 /.075 =, d 6 =,59 /.075 =,868.8

7 Time Series Lecure Noes, 7 6. Perform he proper regression analysis on he deseasonalized daa o obain he appropriae rend model (linear, quadraic, exponenial, ec.). The appropriae rend model for he deseasonalized daa in our example is a linear regression model. The F-es for he coefficien of deerminaion (0.979) and he -es for he slope are boh significan a he 0.05 level (see Table ). The equaion o model he rend is T = An esimae of forecas for any ime period consiss of he produc of he individual componen esimaes a : X = TSC e, Thus, he forecas for he daa in he Table sales in he fourh quarer of he second year ( =8) is X 8 = T8S8C 8 T 7 =, x8 = 75., S 8 = S =.075 C 8 = I (we are assuming here is no cycle) X 8 = 75. x.075 x = Figure : Graph of acual and esimaed values of sales in Table Forecass and Confidence Inervals The esimaes for rend, seasonal variaion, and cycle obained by he muliplicaive decomposiion mehod are used o describe he ime series or o forecas fuure values of he daa. As discussed in an earlier secion, he forecas for ime in a muliplicaive model is he produc of he individual esimaes for ime period. Using our example as shown in Table., sales, we can obain he poin esimae for he second quarer in 988 by he following mehod: X = T S 8 8 8C8 T 8 = x (8) = 96. S 8 =S =.05 C 8 = (we are assuming here is no cycle) X 8 = 0.69 As in he addiive decomposiion mehod, here is no "saisically correc" way o compue a confidence inerval for he poin esimae. This mehod uses he inerval error for he rend componen as he measure of he inerval error of X. Thus, he formula for he confidence inerval can be wrien as xˆ ± α / Se (correcion facor) S where e = he sandard error of esimae ( MSE ) from he appropriae rend regression analysis, and where correcion facor = + ( p ) + n ( ) n where p is he ph ime poin which we concern for confidence inerval, which is obained from he resuls of he regression analysis of he deseasonalized daa. In our example, he rend sandard error for he deseasonalized reail sales is 5.56 and he correcion facor for he 8h ime period is.9. An approximae 95 percen confidence inerval for xˆ 8 would be 0.69 ±.5(5.56)(.9), or (96.57 o 08.80) We can be 95 percen confiden ha he reail sales for he second quarer of 998 will be somewhere beween and (in housands).

8 Time Series Lecure Noes, 8 Tes For Seasonaliy In he preceding examples, we have confirmed he presence of a seasonal componen (before isolaing i) by inspecing he graph of he daa and by prior knowledge of he behavior of he series. There are imes, however, when he presence of a significan seasonal componen is quesionable. In hese insances, somehing more han a visual inspecion of he graph is needed. One such mehod is o apply he Kruskal-Wallis one-way analysis of variance es o he oucomes ha were obained by subracing or dividing he CMAs ino he daa. These oucomes supposed o be conained jus he seasonal and error componens. If here is no specific seasonal componen, he oucomes should consis of nohing bu random error and hus heir disribuion should be he same for all seasons. This means ha if hese oucomes are ranked and he ranks are grouped by seasons, hen he average rank for each season should be saisically equal o he average rank of any oher season. The Kruskal-Wallis es, a nonparameric es analogous o he parameric one-way analysis of variance es, will deermine wheher or no he sums of he rankings (and hus he means) are differen (or he same) beween he various groups (seasons). Table 5: Tesing for seasonaliy: compuaion of he Kruskal-Wallis Saisic for he daa in he Table T Quarer S e Rank Sum of ranks by quarer Hypoheses: : S = S = S = S (here is no seasonaliy) H 0 = H a : Sn for some seasons (here is a seasonaliy in he daa) α = 0.05 This can be accomplished by compuing he saisic R i H = (N + ) N(N + ) n i where N = he oal number of rankings R i = he sum of he rankings in a specific season n i = he number of rankings in a specific season. In he muliplicaive decomposiion example, four seasonal facors (0.90,.05,.005, and.075) were firs isolaed and hen used o deseasonalize he daa. However, if hese four seasonal facors were saisically equal o, hese procedures would no be necessary. By ranking he S e, values and applying he Kruskal-Wallis es o he sums of he seasonal ranks, he presence of he seasonal facors can be saisically confirmed. As seen in Table 5 when he ranks for each quarer are summed and he H saisic is calculaed, he compued value (9.67) is greaer han he abled (criical) value ( χ = 7. 8, df = ). This resul would lead us o conclude ha here is seasonaliy in he reail sales daa in he Table. Tabled (criical) value, χ wih df = L - : χ = 7. 8, df = Compued value: R i H = (N + ) N(N + ) n i 6 7 H = () = 9.67 ()

9 Time Series Lecure Noes, 9 Decision rule: 9.67 > 7.8, herefore rejec H o. Noe, he procedures are he same for esing for seasonaliy in he addiive model- H0 : S = S = S = S = 0. For furher discussion of he Kruskal-Wallis es and oher ess for seasonaliy. Advanages and Disadvanages of he Decomposiion Mehod The main advanages of he decomposiion mehod are he relaive simpliciy of he procedure (i can be accomplished wih a hand calculaor) and he minimal sar-up ime. The disadvanages include no having sound saisical heory behind he mehod, he enire procedure mus be repeaed each ime a new daa poin is acquired, and, as in some oher ime-series echniques, no ouside variables are considered. However, he decomposiion mehod is widely used wih much success and accuracy, especially for shor-erm forecasing. Moving Averages Forecasing If we hen eliminae he firs daa poin from his inerval and add he fourh one, heir moving average is again a forecas for he fifh period. As he mahemaicians would say: x + x + x x N+ M = N where M, is a moving average a he poin, x, is a daa poin in he ime series, and N is he number of daa in he period for which he moving average is calculaed. Therefore, he forecas for he fuure period could be expressed as: x + x + x x N+ F = + N If we agree ha his is a very easy and elegan way of forecasing he fuure, we have o say also ha here are quie a few limiaions o his forecasing mehod. In he firs place his mehod canno forecas he non-saionary series a well enough. Imagine ha we have consanly increasing numbers in he series. Now, having described he rend, we can use moving averages o consruc an equaion ha will simulae our upwards or downwards series. This could be, of course, a simple sraigh-line equaion ha would handle he non saionary ime series. Wha has been menioned above abou every daa poin consising of single and double moving averages, could be convered ino an inercep (or parameer a) of our projeced sraigh line. Therefore: a = M M where M is a single moving average and M is a double moving average. To find he parameer b we have o use he following formula: b = (M M ) N We can now say ha every daa on our ime series could be approximaed by: F = a + b + I is very simple o calculae he forecass of our ime series by saying ha: F + = a + mb Where m is a number of periods ahead, which we are forecasing. m By using he example from Table 6, we can see he mechanics of calculaing forecass. If we compare single and double moving average forecass, hen graphically hey appear as shown in Fig. 5. We can see ha double moving average forecass are following he paern of he original ime series in a no lively fashion, which is exacly wha we waned o achieve in he case of he nonsaionary series.

10 Time Series Lecure Noes, 0 Original, Single and Double Moving Average Forecsa Value Period Figure 5 Single and double moving average forecass Table 6 Single moving averages and double moving average models Period Series M M a b M Forecas = a + b

11 Time Series Lecure Noes, Case Sudy Quesion. (a) Is i possible o apply he decomposiion mehod o Quarerly elecriciy demand in Sri Lanka daa? If so sugges a mehodology. (b) Using he muliplicaive mehod, generae he forecas for he daa as in he Table 7. Table 7: Quarerly elecriciy demand in Sri Lanka for he period of 977/Q o 997/Q 977/ / / / / / / / / / / / / / / / / / / / / Soluion: Table 8 for Calculaed Trend and Seasonal Values under Muliplicaive Model Year Observe Trend Seasonal Year Observer Trend Seasonal &Qr r Values Values &Qr Demand Values Values Demand 977/ / / / / / / / / / /

12 Time Series Lecure Noes, / / / / / / / / / / / demand(mn was) Acual and rend curve of he elecriciy demand in Sri Lanka Quarer Figure 6 Acual and rend values of elecriciy demand daa, Table 9 Calculaed Seasonal Indices for elecriciy demand daa Year Q Q Q Q

13 Time Series Lecure Noes, Toal S-Indices Regression Analysis The regression equaion is T = Predicor Coef SDev T P Consan Elecriciy demand (Mn Was) Acual values versus forecas values of elecriciy demand daa Quarer Figure 7 Acual and esimaed values of elecriciy demand in Sri Lanka Forecass The esimaes for rend, seasonal variaion, and cycle obained by he muliplicaive decomposiion mehod are used o describe he ime series or o forecas fuure values of he daa. As discussed in an earlier secion, he forecas for ime in a muliplicaive model is he produc of he individual esimaes for ime period. Using our example as shown in Table 8 and 9, elecriciy demand in Sri Lanka, we can obain he poin esimae for he second quarer in 998 by he following mehod: X = T S C85 T 85 = x(85)+0.069x(85 ) = S 85 =S =.006 C 85 = (we are assuming here is no cycle) Forecas elecriciy demand a 998/Q is X 85 = Similarly, X 86 = T86S86C86 T 86 = x(86)+0.069x(86 ) = S 86 =S =.0 C 86 = (we are assuming here is no cycle) Forecas elecriciy demand a 998/Q is X 85 =

14 Time Series Lecure Noes, Case sudy Quesion Table 0, shows monhly empirical daa se having seasonaliy. Use appropriae mehod o find adjused seasonal indices. (a) Use muliplicaive decomposing mehod o generae forecass. (b) Explain ineresing feaures in he analysis. Table 0 Monhly daa series for he period of hree years Monh Value Monh Value Monh Value Monh Value Y /Jan Feb March Apr May June July Aug Sep Oc Nov Dec Y /Jan Feb March Apr May June July Aug Sep Oc Nov Dec Y /Jan Feb March Apr May June July Aug Sep Oc Nov Dec Y /Jan Feb March Apr May June July Aug Sep Oc Nov Dec Value Emprical monhy(original) daa series Monh Figure8 Original daa Series as in he Table 0 Regression Analysis: The regression equaion is T = Predicor Coef SDev T P Consan C We have used a ime series ha has monhly values, and ha is four years long. As in he previous example wih he cyclical componen, we have found he rend value of he ime series. If we make a graphic presenaion of wha we have done so far, hen i will appear as shown in Fig. 8.

15 Time Series Lecure Noes, 5 Table Compuaion of seasonal index, by using he muliplicaive decomposiion model of monhly daa series in Table.0 Monh Y /Jan Feb March Apr May June July Aug Sep Oc Nov Dec Y /Jan Feb March Apr May June July Aug Sep Oc Nov Dec X T Inde x S e S e Monh Y Jan Feb March Apr May June July Aug Sep Oc Nov Dec Y /Jan Feb March Apr May June July Aug Sep Oc Nov Dec Table Seasonal Indices of daa in able Adjused monhly Monh Year Year Year Year indices Jan Feb Mar Apr May June July Aug Sep Oc Nov Dec Sum 00 X T Index S e S e

16 Time Series Lecure Noes, 6 Derended daa series as shown in Table 0 Values Monh Figure 9 Derended values of daa in he Table 0 Value Deseasonalised daa series of daa in Table Monh Figure0 Deseasonalised values of daa in he Table 0 Residual Series of daa in he Table 0 Value Monh Figure Residuals(or Irregularaies) of daa in he Table 0 Seasonal Indices of he daa in he Table 9 Seasonal Index Monh Figure Monhly Indices of daa in he Table 0 For he sake of his exercise, if we now divide each acual value wih a responding ypical seasonal index value (see Table 0), we can how he ime series would look if here were no seasonal influences a all. is now jus a deseasonalized value wih some irregular componens presen i (Fig. 0). If we eliminae he irregular componen by dividing he deasonalized values of he series wih he rend, wha we are lef wih is a measure of oher influences or an index of irregular flucuaions (see Table 0). Each value is elling us how much a paricular monh is affeced o he oher influences, and no only seasonal flucuaions. Graphically i appears as shown in Fig.. The adjused seasonal indices as in he Table wih he fuure linear rend values are used o

17 Time Series Lecure Noes, 7 esimae forecas for nex year. This recomposed series, which represens our forecas, is presened in he form of a graph (Fig. ). Acual and Forecased values of daa in he Table.9 values Monh Figure Acual and forecased values of he daa in he Table 0 Obviously, here are several ways of doing wha we have jus described. This migh auomaically imply ha he mehod of classical decomposiion is no paricularly accurae. Forunaely, his is no rue. I is a fairly accurae mehod, bu he problem wih i is ha i is very arbirary, as he resuls can vary depending on who is doing he forecasing. From he classical decomposiion mehod were developed several new modificaions during he 960s and 970s, noe ha hey are jus modernized and compuerized versions of he good old classical decomposiion mehod. Exercises. Consider he ime series daa in Table.Sales of a paricular company, for he period of (ons) Year Q Q Q Q Table.Sales of a paricular company (a) Draw a graph of x() agains and commen upon wheher or daa appears saionary. (b) Calculae a single and double moving average forecas of lengh superimpose boh forecass on o your graph. (c) Calculae he forecas for quarer, 980 using he double moving and compare his resul wih your resul from decomposiion mehod.. Year Q Q Q Q Table Sales of heaing oil. (a) Draw a graph of x() agains and commen upon wheher or no he daa is saionary. (b) Superimpose on o he above graph a moving average forecas of suiable lengh. Give reasons for your model choice. (c) Use his moving average o provide a forecas for he firs quarer of year.. Table 5 gives he index of average earnings of insurance, banking and finance employees (base: 976 = 00). Year March June Sep: Dec:

18 Time Series Lecure Noes, (a) Draw a graph of x() agains and superimpose on he graph moving averages of lengh in order o show he general rend. (b) Provide an index forecas for he four quarers of 98.. Year Q Q Q Q Table 6: Number of cans (in 00000s) of lager sold manufacurer in each quarer of hree successive years. (a) Draw a graph of x() agains and commen upon wheher or no he daa is saionary. (b) Superimpose ono he above graph a moving average forecas of suiable lengh. Give reasons for your model choice. (c) Use his moving average o provide a forecas for he firs quarer of Table 7 shows he number of cans (in 00000s) of lager sold by a manufacurer in each quarer of hree successive years. Table 6 Year Q Q Q Q (a) Draw a graph of x() agains. (b) Compue he rend equaion using linear regression analysis. (c) Wha are he differences beween an `addiive' and `muliplicaive models when calculaing he seasonal componens? (d) Esimae he number of cans sold in each quarer of 988 using boh he muliplicaive (A= T x S) and `addiive' (A = T + S) models. 6. Use he daa in he able., relae o he sales of heaing oil. (a) Draw a graph of x() agains. (b) Compue he rend equaion using linear regression analysis. (c) Use he Muliplicaive model, A = T x S, o provide a forecas for he firs quarer of year. 7. Table 8 below gives he index of average earnings (GB) of insurance, banking and finance employees (base: 976 = 00). Table 8 Year Q Q Q Q (a) Draw a graph of x() agains. ` (b) Compue he rend equaion using linear regression analysis. (c) Provide an index forecas for Q, Q and Q in he year. Give reasons for your model choice.

The naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1

The naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1 Business Condiions & Forecasing Exponenial Smoohing LECTURE 2 MOVING AVERAGES AND EXPONENTIAL SMOOTHING OVERVIEW This lecure inroduces ime-series smoohing forecasing mehods. Various models are discussed,

More information

Chapter 8 Student Lecture Notes 8-1

Chapter 8 Student Lecture Notes 8-1 Chaper Suden Lecure Noes - Chaper Goals QM: Business Saisics Chaper Analyzing and Forecasing -Series Daa Afer compleing his chaper, you should be able o: Idenify he componens presen in a ime series Develop

More information

Usefulness of the Forward Curve in Forecasting Oil Prices

Usefulness of the Forward Curve in Forecasting Oil Prices Usefulness of he Forward Curve in Forecasing Oil Prices Akira Yanagisawa Leader Energy Demand, Supply and Forecas Analysis Group The Energy Daa and Modelling Cener Summary When people analyse oil prices,

More information

Economics 140A Hypothesis Testing in Regression Models

Economics 140A Hypothesis Testing in Regression Models Economics 140A Hypohesis Tesing in Regression Models While i is algebraically simple o work wih a populaion model wih a single varying regressor, mos populaion models have muliple varying regressors 1

More information

Chapter 8: Regression with Lagged Explanatory Variables

Chapter 8: Regression with Lagged Explanatory Variables Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One

More information

Vector Autoregressions (VARs): Operational Perspectives

Vector Autoregressions (VARs): Operational Perspectives Vecor Auoregressions (VARs): Operaional Perspecives Primary Source: Sock, James H., and Mark W. Wason, Vecor Auoregressions, Journal of Economic Perspecives, Vol. 15 No. 4 (Fall 2001), 101-115. Macroeconomericians

More information

INTRODUCTION TO FORECASTING

INTRODUCTION TO FORECASTING INTRODUCTION TO FORECASTING INTRODUCTION: Wha is a forecas? Why do managers need o forecas? A forecas is an esimae of uncerain fuure evens (lierally, o "cas forward" by exrapolaing from pas and curren

More information

Graphing the Von Bertalanffy Growth Equation

Graphing the Von Bertalanffy Growth Equation file: d:\b173-2013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and

More information

Revisions to Nonfarm Payroll Employment: 1964 to 2011

Revisions to Nonfarm Payroll Employment: 1964 to 2011 Revisions o Nonfarm Payroll Employmen: 1964 o 2011 Tom Sark December 2011 Summary Over recen monhs, he Bureau of Labor Saisics (BLS) has revised upward is iniial esimaes of he monhly change in nonfarm

More information

House Price Index (HPI)

House Price Index (HPI) House Price Index (HPI) The price index of second hand houses in Colombia (HPI), regisers annually and quarerly he evoluion of prices of his ype of dwelling. The calculaion is based on he repeaed sales

More information

Cointegration: The Engle and Granger approach

Cointegration: The Engle and Granger approach Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be non-saionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require

More information

Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that

Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex

More information

An empirical analysis about forecasting Tmall air-conditioning sales using time series model Yan Xia

An empirical analysis about forecasting Tmall air-conditioning sales using time series model Yan Xia An empirical analysis abou forecasing Tmall air-condiioning sales using ime series model Yan Xia Deparmen of Mahemaics, Ocean Universiy of China, China Absrac Time series model is a hospo in he research

More information

Chapter 6: Business Valuation (Income Approach)

Chapter 6: Business Valuation (Income Approach) Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he

More information

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches. Appendi A: Area worked-ou s o Odd-Numbered Eercises Do no read hese worked-ou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa

More information

Part 1: White Noise and Moving Average Models

Part 1: White Noise and Moving Average Models Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical

More information

DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations

DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations DIFFERENTIAL EQUATIONS wih TI-89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows

More information

1. The graph shows the variation with time t of the velocity v of an object.

1. The graph shows the variation with time t of the velocity v of an object. 1. he graph shows he variaion wih ime of he velociy v of an objec. v Which one of he following graphs bes represens he variaion wih ime of he acceleraion a of he objec? A. a B. a C. a D. a 2. A ball, iniially

More information

Morningstar Investor Return

Morningstar Investor Return Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion

More information

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1 Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

More information

RC, RL and RLC circuits

RC, RL and RLC circuits Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.

More information

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion

More information

Research Question Is the average body temperature of healthy adults 98.6 F? Introduction to Hypothesis Testing. Statistical Hypothesis

Research Question Is the average body temperature of healthy adults 98.6 F? Introduction to Hypothesis Testing. Statistical Hypothesis Inroducion o Hypohesis Tesing Research Quesion Is he average body emperaure of healhy aduls 98.6 F? HT - 1 HT - 2 Scienific Mehod 1. Sae research hypoheses or quesions. µ = 98.6? 2. Gaher daa or evidence

More information

SEASONAL ADJUSTMENT. 1 Introduction. 2 Methodology. 3 X-11-ARIMA and X-12-ARIMA Methods

SEASONAL ADJUSTMENT. 1 Introduction. 2 Methodology. 3 X-11-ARIMA and X-12-ARIMA Methods SEASONAL ADJUSTMENT 1 Inroducion 2 Mehodology 2.1 Time Series and Is Componens 2.1.1 Seasonaliy 2.1.2 Trend-Cycle 2.1.3 Irregulariy 2.1.4 Trading Day and Fesival Effecs 3 X-11-ARIMA and X-12-ARIMA Mehods

More information

Journal Of Business & Economics Research September 2005 Volume 3, Number 9

Journal Of Business & Economics Research September 2005 Volume 3, Number 9 Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy Yi-Kang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo

More information

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

More information

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment. . Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure

More information

Math 201 Lecture 12: Cauchy-Euler Equations

Math 201 Lecture 12: Cauchy-Euler Equations Mah 20 Lecure 2: Cauchy-Euler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem

More information

Fourier Series Solution of the Heat Equation

Fourier Series Solution of the Heat Equation Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,

More information

The Transport Equation

The Transport Equation The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

More information

Stability. Coefficients may change over time. Evolution of the economy Policy changes

Stability. Coefficients may change over time. Evolution of the economy Policy changes Sabiliy Coefficiens may change over ime Evoluion of he economy Policy changes Time Varying Parameers y = α + x β + Coefficiens depend on he ime period If he coefficiens vary randomly and are unpredicable,

More information

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins) Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer

More information

Relative velocity in one dimension

Relative velocity in one dimension Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies

More information

Hedging with Forwards and Futures

Hedging with Forwards and Futures Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buy-side of a forward/fuures

More information

Rotational Inertia of a Point Mass

Rotational Inertia of a Point Mass Roaional Ineria of a Poin Mass Saddleback College Physics Deparmen, adaped from PASCO Scienific PURPOSE The purpose of his experimen is o find he roaional ineria of a poin experimenally and o verify ha

More information

Why Do Real and Nominal. Inventory-Sales Ratios Have Different Trends?

Why Do Real and Nominal. Inventory-Sales Ratios Have Different Trends? Why Do Real and Nominal Invenory-Sales Raios Have Differen Trends? By Valerie A. Ramey Professor of Economics Deparmen of Economics Universiy of California, San Diego and Research Associae Naional Bureau

More information

4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay

4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay 324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find

More information

Time Series Analysis Using SAS R Part I The Augmented Dickey-Fuller (ADF) Test

Time Series Analysis Using SAS R Part I The Augmented Dickey-Fuller (ADF) Test ABSTRACT Time Series Analysis Using SAS R Par I The Augmened Dickey-Fuller (ADF) Tes By Ismail E. Mohamed The purpose of his series of aricles is o discuss SAS programming echniques specifically designed

More information

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur Module 4 Single-phase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,

More information

Issues Using OLS with Time Series Data. Time series data NOT randomly sampled in same way as cross sectional each obs not i.i.d

Issues Using OLS with Time Series Data. Time series data NOT randomly sampled in same way as cross sectional each obs not i.i.d These noes largely concern auocorrelaion Issues Using OLS wih Time Series Daa Recall main poins from Chaper 10: Time series daa NOT randomly sampled in same way as cross secional each obs no i.i.d Why?

More information

Chapter 4. Properties of the Least Squares Estimators. Assumptions of the Simple Linear Regression Model. SR3. var(e t ) = σ 2 = var(y t )

Chapter 4. Properties of the Least Squares Estimators. Assumptions of the Simple Linear Regression Model. SR3. var(e t ) = σ 2 = var(y t ) Chaper 4 Properies of he Leas Squares Esimaors Assumpions of he Simple Linear Regression Model SR1. SR. y = β 1 + β x + e E(e ) = 0 E[y ] = β 1 + β x SR3. var(e ) = σ = var(y ) SR4. cov(e i, e j ) = cov(y

More information

Forecasting, Ordering and Stock- Holding for Erratic Demand

Forecasting, Ordering and Stock- Holding for Erratic Demand ISF 2002 23 rd o 26 h June 2002 Forecasing, Ordering and Sock- Holding for Erraic Demand Andrew Eaves Lancaser Universiy / Andalus Soluions Limied Inroducion Erraic and slow-moving demand Demand classificaion

More information

11. Tire pressure. Here we always work with relative pressure. That s what everybody always does.

11. Tire pressure. Here we always work with relative pressure. That s what everybody always does. 11. Tire pressure. The graph You have a hole in your ire. You pump i up o P=400 kilopascals (kpa) and over he nex few hours i goes down ill he ire is quie fla. Draw wha you hink he graph of ire pressure

More information

Principal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.

Principal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya. Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one

More information

Acceleration Lab Teacher s Guide

Acceleration Lab Teacher s Guide Acceleraion Lab Teacher s Guide Objecives:. Use graphs of disance vs. ime and velociy vs. ime o find acceleraion of a oy car.. Observe he relaionship beween he angle of an inclined plane and he acceleraion

More information

Forecasting. Including an Introduction to Forecasting using the SAP R/3 System

Forecasting. Including an Introduction to Forecasting using the SAP R/3 System Forecasing Including an Inroducion o Forecasing using he SAP R/3 Sysem by James D. Blocher Vincen A. Maber Ashok K. Soni Munirpallam A. Venkaaramanan Indiana Universiy Kelley School of Business February

More information

Measuring macroeconomic volatility Applications to export revenue data, 1970-2005

Measuring macroeconomic volatility Applications to export revenue data, 1970-2005 FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970-005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a

More information

HANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed.

HANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed. Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can

More information

CHARGE AND DISCHARGE OF A CAPACITOR

CHARGE AND DISCHARGE OF A CAPACITOR REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:

More information

COMPARISON OF AIR TRAVEL DEMAND FORECASTING METHODS

COMPARISON OF AIR TRAVEL DEMAND FORECASTING METHODS COMPARISON OF AIR RAVE DEMAND FORECASING MEHODS Ružica Škurla Babić, M.Sc. Ivan Grgurević, B.Eng. Universiy of Zagreb Faculy of ranspor and raffic Sciences Vukelićeva 4, HR- Zagreb, Croaia skurla@fpz.hr,

More information

Chapter 2 Kinematics in One Dimension

Chapter 2 Kinematics in One Dimension Chaper Kinemaics in One Dimension Chaper DESCRIBING MOTION:KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings moe how far (disance and displacemen), how fas (speed and elociy), and how

More information

Return Calculation of U.S. Treasury Constant Maturity Indices

Return Calculation of U.S. Treasury Constant Maturity Indices Reurn Calculaion of US Treasur Consan Mauri Indices Morningsar Mehodolog Paper Sepeber 30 008 008 Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion

More information

Chapter 2: Principles of steady-state converter analysis

Chapter 2: Principles of steady-state converter analysis Chaper 2 Principles of Seady-Sae Converer Analysis 2.1. Inroducion 2.2. Inducor vol-second balance, capacior charge balance, and he small ripple approximaion 2.3. Boos converer example 2.4. Cuk converer

More information

State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University

State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween

More information

Technical Description of S&P 500 Buy-Write Monthly Index Composition

Technical Description of S&P 500 Buy-Write Monthly Index Composition Technical Descripion of S&P 500 Buy-Wrie Monhly Index Composiion The S&P 500 Buy-Wrie Monhly (BWM) index is a oal reurn index based on wriing he nearby a-he-money S&P 500 call opion agains he S&P 500 index

More information

Understanding Sequential Circuit Timing

Understanding Sequential Circuit Timing ENGIN112: Inroducion o Elecrical and Compuer Engineering Fall 2003 Prof. Russell Tessier Undersanding Sequenial Circui Timing Perhaps he wo mos disinguishing characerisics of a compuer are is processor

More information

A Mathematical Description of MOSFET Behavior

A Mathematical Description of MOSFET Behavior 10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical

More information

4 Convolution. Recommended Problems. x2[n] 1 2[n]

4 Convolution. Recommended Problems. x2[n] 1 2[n] 4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discree-ime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.1-1.

More information

SPECIAL REPORT May 4, Shifting Drivers of Inflation Canada versus the U.S.

SPECIAL REPORT May 4, Shifting Drivers of Inflation Canada versus the U.S. Paul Ferley Assisan Chief Economis 416-974-7231 paul.ferley@rbc.com Nahan Janzen Economis 416-974-0579 nahan.janzen@rbc.com SPECIAL REPORT May 4, 2010 Shifing Drivers of Inflaion Canada versus he U.S.

More information

Two Compartment Body Model and V d Terms by Jeff Stark

Two Compartment Body Model and V d Terms by Jeff Stark Two Comparmen Body Model and V d Terms by Jeff Sark In a one-comparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics - By his, we mean ha eliminaion is firs order and ha pharmacokineic

More information

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613. Graduae School of Business Adminisraion Universiy of Virginia UVA-F-38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised

More information

II.1. Debt reduction and fiscal multipliers. dbt da dpbal da dg. bal

II.1. Debt reduction and fiscal multipliers. dbt da dpbal da dg. bal Quarerly Repor on he Euro Area 3/202 II.. Deb reducion and fiscal mulipliers The deerioraion of public finances in he firs years of he crisis has led mos Member Saes o adop sizeable consolidaion packages.

More information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

More information

Multiple Structural Breaks in the Nominal Interest Rate and Inflation in Canada and the United States

Multiple Structural Breaks in the Nominal Interest Rate and Inflation in Canada and the United States Deparmen of Economics Discussion Paper 00-07 Muliple Srucural Breaks in he Nominal Ineres Rae and Inflaion in Canada and he Unied Saes Frank J. Akins, Universiy of Calgary Preliminary Draf February, 00

More information

Forecasting Malaysian Gold Using. GARCH Model

Forecasting Malaysian Gold Using. GARCH Model Applied Mahemaical Sciences, Vol. 7, 2013, no. 58, 2879-2884 HIKARI Ld, www.m-hikari.com Forecasing Malaysian Gold Using GARCH Model Pung Yean Ping 1, Nor Hamizah Miswan 2 and Maizah Hura Ahmad 3 Deparmen

More information

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,

More information

A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)

A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM) A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke

More information

Journal of Business & Economics Research Volume 1, Number 10

Journal of Business & Economics Research Volume 1, Number 10 Annualized Invenory/Sales Journal of Business & Economics Research Volume 1, Number 1 A Macroeconomic Analysis Of Invenory/Sales Raios William M. Bassin, Shippensburg Universiy Michael T. Marsh (E-mail:

More information

Inventory Planning with Forecast Updates: Approximate Solutions and Cost Error Bounds

Inventory Planning with Forecast Updates: Approximate Solutions and Cost Error Bounds OPERATIONS RESEARCH Vol. 54, No. 6, November December 2006, pp. 1079 1097 issn 0030-364X eissn 1526-5463 06 5406 1079 informs doi 10.1287/opre.1060.0338 2006 INFORMS Invenory Planning wih Forecas Updaes:

More information

MTH6121 Introduction to Mathematical Finance Lesson 5

MTH6121 Introduction to Mathematical Finance Lesson 5 26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random

More information

Chabot College Physics Lab RC Circuits Scott Hildreth

Chabot College Physics Lab RC Circuits Scott Hildreth Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard

More information

3 Runge-Kutta Methods

3 Runge-Kutta Methods 3 Runge-Kua Mehods In conras o he mulisep mehods of he previous secion, Runge-Kua mehods are single-sep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he

More information

YEN FUTURES: EXAMINING HEDGING EFFECTIVENESS BIAS AND CROSS-CURRENCY HEDGING RESULTS ROBERT T. DAIGLER FLORIDA INTERNATIONAL UNIVERSITY SUBMITTED FOR

YEN FUTURES: EXAMINING HEDGING EFFECTIVENESS BIAS AND CROSS-CURRENCY HEDGING RESULTS ROBERT T. DAIGLER FLORIDA INTERNATIONAL UNIVERSITY SUBMITTED FOR YEN FUTURES: EXAMINING HEDGING EFFECTIVENESS BIAS AND CROSS-CURRENCY HEDGING RESULTS ROBERT T. DAIGLER FLORIDA INTERNATIONAL UNIVERSITY SUBMITTED FOR THE FIRST ANNUAL PACIFIC-BASIN FINANCE CONFERENCE The

More information

MACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR

MACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR MACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR The firs experimenal publicaion, which summarised pas and expeced fuure developmen of basic economic indicaors, was published by he Minisry

More information

Hotel Room Demand Forecasting via Observed Reservation Information

Hotel Room Demand Forecasting via Observed Reservation Information Proceedings of he Asia Pacific Indusrial Engineering & Managemen Sysems Conference 0 V. Kachivichyanuul, H.T. Luong, and R. Piaaso Eds. Hoel Room Demand Forecasing via Observed Reservaion Informaion aragain

More information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

More information

Classification based Expert Selection for Accurate Sales Forecasting

Classification based Expert Selection for Accurate Sales Forecasting Inernaional Journal of Compuer Applicaions (0975 8887) Classificaion based Exper Selecion for Accurae Sales Forecasing Darshana D. Chande Compuer Engineering Deparmen, Governmen polyechnic, Thane M.Vijayalakshmi

More information

Small and Large Trades Around Earnings Announcements: Does Trading Behavior Explain Post-Earnings-Announcement Drift?

Small and Large Trades Around Earnings Announcements: Does Trading Behavior Explain Post-Earnings-Announcement Drift? Small and Large Trades Around Earnings Announcemens: Does Trading Behavior Explain Pos-Earnings-Announcemen Drif? Devin Shanhikumar * Firs Draf: Ocober, 2002 This Version: Augus 19, 2004 Absrac This paper

More information

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

More information

Why Did the Demand for Cash Decrease Recently in Korea?

Why Did the Demand for Cash Decrease Recently in Korea? Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in

More information

17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides

17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides 7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion

More information

Cointegration Analysis of Exchange Rate in Foreign Exchange Market

Cointegration Analysis of Exchange Rate in Foreign Exchange Market Coinegraion Analysis of Exchange Rae in Foreign Exchange Marke Wang Jian, Wang Shu-li School of Economics, Wuhan Universiy of Technology, P.R.China, 430074 Absrac: This paper educed ha he series of exchange

More information

Fourier series. Learning outcomes

Fourier series. Learning outcomes Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Half-range series 6. The complex form 7. Applicaion of Fourier series

More information

Information Theoretic Evaluation of Change Prediction Models for Large-Scale Software

Information Theoretic Evaluation of Change Prediction Models for Large-Scale Software Informaion Theoreic Evaluaion of Change Predicion Models for Large-Scale Sofware Mina Askari School of Compuer Science Universiy of Waerloo Waerloo, Canada maskari@uwaerloo.ca Ric Hol School of Compuer

More information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

More information

COMPUTATION OF CENTILES AND Z-SCORES FOR HEIGHT-FOR-AGE, WEIGHT-FOR-AGE AND BMI-FOR-AGE

COMPUTATION OF CENTILES AND Z-SCORES FOR HEIGHT-FOR-AGE, WEIGHT-FOR-AGE AND BMI-FOR-AGE COMPUTATION OF CENTILES AND Z-SCORES FOR HEIGHT-FOR-AGE, WEIGHT-FOR-AGE AND BMI-FOR-AGE The mehod used o consruc he 2007 WHO references relied on GAMLSS wih he Box-Cox power exponenial disribuion (Rigby

More information

Signal Rectification

Signal Rectification 9/3/25 Signal Recificaion.doc / Signal Recificaion n imporan applicaion of juncion diodes is signal recificaion. here are wo ypes of signal recifiers, half-wae and fullwae. Le s firs consider he ideal

More information

Lecture III: Finish Discounted Value Formulation

Lecture III: Finish Discounted Value Formulation Lecure III: Finish Discouned Value Formulaion I. Inernal Rae of Reurn A. Formally defined: Inernal Rae of Reurn is ha ineres rae which reduces he ne presen value of an invesmen o zero.. Finding he inernal

More information

Chapter 1.6 Financial Management

Chapter 1.6 Financial Management Chaper 1.6 Financial Managemen Par I: Objecive ype quesions and answers 1. Simple pay back period is equal o: a) Raio of Firs cos/ne yearly savings b) Raio of Annual gross cash flow/capial cos n c) = (1

More information

Permutations and Combinations

Permutations and Combinations Permuaions and Combinaions Combinaorics Copyrigh Sandards 006, Tes - ANSWERS Barry Mabillard. 0 www.mah0s.com 1. Deermine he middle erm in he expansion of ( a b) To ge he k-value for he middle erm, divide

More information

The Relation Between T-score, Z-score, Bone Mineral Density and Body Mass Index

The Relation Between T-score, Z-score, Bone Mineral Density and Body Mass Index The Relaion Beween T-score, Z-score, Bone Mineral Densiy and Body Mass Index RABA'A KAREEM FARES AL-MAITAH AL-Balqa Applied Universiy (JORDAN) Tel: 00962-796676697 *E-mail:hamzaalawi@ymail.com Absrac:

More information

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,

More information

Diagnostic Examination

Diagnostic Examination Diagnosic Examinaion TOPIC XV: ENGINEERING ECONOMICS TIME LIMIT: 45 MINUTES 1. Approximaely how many years will i ake o double an invesmen a a 6% effecive annual rae? (A) 10 yr (B) 12 yr (C) 15 yr (D)

More information

4. International Parity Conditions

4. International Parity Conditions 4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency

More information

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes

More information

WATER MIST FIRE PROTECTION RELIABILITY ANALYSIS

WATER MIST FIRE PROTECTION RELIABILITY ANALYSIS WATER MIST FIRE PROTECTION RELIABILITY ANALYSIS Shuzhen Xu Research Risk and Reliabiliy Area FM Global Norwood, Massachuses 262, USA David Fuller Engineering Sandards FM Global Norwood, Massachuses 262,

More information

Newton s Laws of Motion

Newton s Laws of Motion Newon s Laws of Moion MS4414 Theoreical Mechanics Firs Law velociy. In he absence of exernal forces, a body moves in a sraigh line wih consan F = 0 = v = cons. Khan Academy Newon I. Second Law body. The

More information

Supply Chain Management Using Simulation Optimization By Miheer Kulkarni

Supply Chain Management Using Simulation Optimization By Miheer Kulkarni Supply Chain Managemen Using Simulaion Opimizaion By Miheer Kulkarni This problem was inspired by he paper by Jung, Blau, Pekny, Reklaii and Eversdyk which deals wih supply chain managemen for he chemical

More information

Differential Equations and Linear Superposition

Differential Equations and Linear Superposition Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y

More information

AP Calculus BC 2010 Scoring Guidelines

AP Calculus BC 2010 Scoring Guidelines AP Calculus BC Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board

More information